Question: Let $S$ be a surface in 3D described by the equation $z = 6 - 2x^2 - 3y^2$. Fill in the rest of the equation of the plane tangent to $S$ at $(3, 1)$. $z = $
Solution: The equation for a tangent plane of an explicitly defined surface $z = f(x, y)$ at the point $(a, b)$ is: $f(a, b) + f_x(x - a) + f_y(y - b) = z$ [What's the intuition behind the formula?] We can see from the formula that the two values we're missing are $f(3, 1)$ and $f_x$. $\begin{aligned} &f(3, 1) = 6 - 18 - 3 = -15 \\ \\ &f_x = -4x = -12 \end{aligned}$ Here's the completed equation for the tangent plane of $S$ at $(3, 1)$ : $z = -15 - 12(x - 3) - 6(y - 1)$